# 时间序列的自回归模型—从线性代数的角度来看

### Fibonacci 序列

$F_{n}=F_{n-1}+F_{n-2}$

0，1，1，2，3，5，8，13，21，34，55，89，144，…

### 求解 Fibonacci 序列的通项公式 －－－ 矩阵对角化

$\left( \begin{array}{c} F_{n+2} \\ F_{n+1} \\ \end{array}\right)= \left( \begin{array}{cc} 1 & 1 \\ 1 & 0 \\ \end{array}\right) \left( \begin{array}{c} F_{n+1} \\ F_{n} \\ \end{array}\right) = A \left( \begin{array}{c} F_{n+1} \\ F_{n} \\ \end{array}\right),$

$\left( \begin{array}{c} F_{n} \\ F_{n-1} \\ \end{array}\right)= A \left( \begin{array}{c} F_{n-1} \\ F_{n-2} \\ \end{array}\right) = \cdots = A^{n-1} \left( \begin{array}{c} F_{1} \\ F_{0} \\ \end{array}\right).$

$P^{-1}AP = diag(\lambda_{1},\cdots,\lambda_{m})$

$\implies AP = P diag(\lambda_{1},\cdots,\lambda_{m})$

$\implies A = P diag(\lambda_{1},\cdots,\lambda_{m})P^{-1}.$

$A^{k} = (PDP^{-1})\cdots(PDP^{-1}) = P D^{k} P^{-1}= P diag(\lambda_{1}^{k},\cdots,\lambda_{m}^{k})P^{-1}.$

$det(\lambda I - A) = 0,$

$A = \left( \begin{array}{cc} 1 & 1 \\ 1 & 0 \\ \end{array}\right)$,

$\vec{\alpha}_{1} = (\phi,1)^{T}, \vec{\alpha}_{2} = (-\phi^{-1},1)$.

$F_{k} = \frac{1}{\sqrt{5}}\bigg(\frac{1+\sqrt{5}}{2}\bigg)^{k} - \frac{1}{\sqrt{5}}\bigg(\frac{1-\sqrt{5}}{2}\bigg)^{k}=\frac{\phi^{k}-(-\phi)^{-k}}{\sqrt{5}}$.

### 时间序列的弱平稳性

1. $E(x_{t})$ 对于所有的 $t\geq 0$ 都是恒定的；
2. $Var(x_{t})$ 对于所有的 $t\geq 0$ 都是恒定的；
3. $x_{t}$$x_{t-h}$ 的协方差对于所有的 $t\geq 0$ 都是恒定的。

$ACF(x_{t},x_{t-h}) = \frac{Covariance(x_{t},x_{t-h})}{\sqrt{Var(x_{t})\cdot Var(x_{t-h})}}$.

$ACF(x_{t},x_{t-h}) = \frac{Covariance(x_{t},x_{t-h})}{Var(x_{t})}$.

### 时间序列的自回归模型（AutoRegression Model）

#### AR(1) 模型

AR(1) 模型指的是时间序列 $\{x_{t}\}_{t\geq 0}$ 在时间戳 $t$ 时刻的取值 $x_{t}$ 与时间戳 $t - 1$ 时刻的取值 $x_{t-1}$ 相关，其公式就是：

$x_{t}=\delta+\phi_{1}x_{t-1}+w_{t}$

1. $w_{t}\sim N(0,\sigma_{w}^{2})$，并且 $w_{t}$ 满足 iid 条件。其中 $N(0,\sigma_{w}^{2})$ 表示 Gauss 正态分布，它的均值是0，方差是 $\sigma_{w}^{2}$
2. $w_{t}$$x_{t}$ 是相互独立的（independent）。
3. $x_{0},x_{1},\cdots$弱平稳的，i.e. 必须满足 $|\phi_{1}|<1$

1. $E(x_{t}) = \delta/(1-\phi_{1})$.
2. $Var(x_{t}) = \sigma_{w}^{2}/(1-\phi_{1}^{2})$.
3. $Covariance(x_{t},x_{t-h}) = \phi_{1}^{h}$.

Proof of 1. 从 AR(1) 的模型出发，可以得到

$E(x_{t}) = E(\delta + \phi_{1}x_{t-1}+w_{t}) = \delta + \phi_{1}E(x_{t-1}) = \delta + \phi_{1}E(x_{t})$,

Proof of 2. 从 AR(1) 的模型出发，可以得到

$Var(x_{t}) = Var(\delta + \phi_{1}x_{t-1}+w_{t})$

$= \phi_{1}^{2}Var(x_{t-1}) +Var(w_{t}) = \phi_{1}^{2}Var(x_{t}) + \sigma_{w}^{2}$,

Proof of 3.$\mu = E(x_{t}), \forall t\geq 0$. 从 $x_{t}$ 的定义出发，可以得到：

$x_{t}-\mu = \phi_{1}(x_{t-1}-\mu)+w_{t}$

$= \phi_{1}^{h}(x_{t-h}-\mu) + \phi_{1}^{h-1}w_{t-h+1}+\cdots+\phi_{1}w_{t-1}+w_{t},$

$\rho_{h} = Covariance(x_{t},x_{t-h}) = \frac{E((x_{t}-\mu)\cdot(x_{t-h}-\mu))}{Var(x_{t})}=\phi_{1}^{h}$.

#### AR(1) 模型与一维动力系统

$f(x) = \phi_{1}x + \delta,$

Method 1.

$f^{n}(x) = \phi_{1}^{n}x+ \frac{1-\phi_{1}^{n}}{1-\phi_{1}}\delta$,

$n\rightarrow \infty$，可以得到 $f^{n}(x)\rightarrow \delta/(1-\phi_{1})$。这与 $E(x_{t}) = \delta/(1-\phi_{1})$ 其实是保持一致的。

Method 2.

$f(x)-\frac{\delta}{1-\phi_{1}} = \phi_{1}(x-\frac{\delta}{1-\phi_{1}})$

$\implies |f(x)-\frac{\delta}{1-\phi_{1}}| <\frac{1+|\phi_{1}|}{2}\cdot|x-\frac{\delta}{1-\phi_{1}}|$

$\implies |f^{n}(x)-\frac{\delta}{1-\phi_{1}}|<\bigg(\frac{1+|\phi_{1}|}{2}\bigg)^{n}\cdot|x-\frac{\delta}{1-\phi_{1}}|$.

$f(x)-\frac{\delta}{1-\phi_{1}} = \phi_{1}(x-\frac{\delta}{1-\phi_{1}})$

$\implies |f(x)-\frac{\delta}{1-\phi_{1}}| >\frac{1+|\phi_{1}|}{2}\cdot|x-\frac{\delta}{1-\phi_{1}}|$

$\implies |f^{n}(x)-\frac{\delta}{1-\phi_{1}}|>\bigg(\frac{1+|\phi_{1}|}{2}\bigg)^{n}\cdot|x-\frac{\delta}{1-\phi_{1}}|$.

#### AR(p) 模型

1. AR(1) 模型形如：

$x_{t}=\delta+\phi_{1}x_{t-1}+w_{t}.$

2. AR(2) 模型形如：

$x_{t} = \delta + \phi_{1}x_{t-1}+\phi_{2}x_{t-2}+w_{t}.$

3. AR(p) 模型形如：

$x_{t} = \delta + \phi_{1}x_{t-1}+\phi_{2}x_{t-2}+\cdots+\phi_{p}x_{t-p}+w_{t}.$

#### AR(p) 模型的稳定性 －－－ 基于线性代数

$x_{t}= \phi_{1}x_{t-1} + \phi_{2}x_{t-2}$.

$\left( \begin{array}{c} x_{t+2} \\ x_{t+1} \\ \end{array}\right)= \left( \begin{array}{cc} \phi_{1} & \phi_{2} \\ 1 & 0 \\ \end{array}\right) \left( \begin{array}{c} x_{t+1} \\ x_{t} \\ \end{array}\right) = A \left( \begin{array}{c} x_{t+1} \\ x_{t} \\ \end{array}\right).$

$x_{t} = \phi_{1}x_{t-1}+\phi_{2}x_{t-2}+\cdots+\phi_{p}x_{t-p}.$

$\left(\begin{array}{c} x_{t+p} \\ x_{t+p-1} \\ \vdots \\ x_{t+1}\\ \end{array}\right) = \left(\begin{array}{ccccc} \phi_{1} & \phi_{2} &\cdots & \phi_{p-1} & \phi_{p} \\ 1 & 0 & \cdots & 0 & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \cdots & 1 & 0 \\ \end{array}\right) \left(\begin{array}{c} x_{t+p-1} \\ x_{t+p-2} \\ \vdots \\ x_{t} \\ \end{array}\right) = A \left(\begin{array}{c} x_{t+p-1} \\ x_{t+p-2} \\ \vdots \\ x_{t} \\ \end{array}\right)$

$\lambda^{p}-\phi_{1}\lambda^{p-1}-\phi_{2}\lambda^{p-2}-\cdots-\phi_{p}=0.$

$x_{t} = \phi_{1}x_{t-1}+\phi_{2}x_{t-2}+\cdots+\phi_{p}x_{t-p}$

# 如何理解时间序列？— 从Riemann积分和Lebesgue积分谈起

Riemann 积分和 Lebesgue 积分是数学中两个非常重要的概念。本文将会从 Riemann 积分和 Lebesgue 积分的定义出发，介绍它们各自的性质和联系。

## 积分

### Riemann 积分

Riemann 积分虽然被称为 Riemann 积分，但是在 Riemann 之前就有学者对这类积分进行了详细的研究。早在阿基米德时代，阿基米德为了计算曲线 $x^{2}$ 在 [0,1] 区间上与 X 坐标轴所夹的图形面积，就使用了 Riemann 积分的思想。 他把 [0,1] 区间等长地切割成 n 段，每一段使用一个长方形去逼近 $x^{2}$ 这条曲线的分段面积，再把 n 取得很大，所以得到当 n 趋近于无穷的时候，就知道该面积其实是 1/3。

$\sum_{i=0}^{n-1}f(t_{i})(x_{i+1}-x_{i}).$

$|\sum_{i=0}^{n-1}f(t_{i})(x_{i+1}-x_{i}) - s|<\epsilon.$

### Lebesgue 积分

Riemann 积分是为了计算曲线与 X 轴所围成的面积，而 Lebesgue 积分也是做同样的事情，但是计算面积的方法略有不同。要想直观的解释两种积分的原理，可以参见下图：

Riemann 积分是把一条曲线的底部分成等长的区间，测量每一个区间上的曲线高度，所以总面积就是这些区间与高度所围成的面积和。

Lebesgue 积分是把曲线化成等高线图，每两根相邻等高线的差值是一样的。每根等高线之内含有它所圈着的长度，因此总面积就是这些等高线内的面积之和。

1. Riemann 积分：从一个角落开始一口一口吃，每口都包含所有的配料；
2. Lebesgue 积分：从最上层开始吃，按照“面包-配菜-肉-蛋-面包”的节奏，一层一层来吃。

$\int(\sum_{k}a_{k}1_{S_{k}})d\mu = \sum_{k}a_{k}\int 1_{S_{k}}d\mu = \sum_{k}a_{k}\mu(S_{k}).$

$\int_{E}f d\mu = \sup\{\int_{E}sd\mu: \bold{0}\leq s\leq f\}$,

$\int fd\mu = \int f^{+}d\mu - \int f^{-}d\mu.$.

### Riemann 积分与Lebesgue 积分的关系

$(R)\int_{a}^{b}f(x)dx = (L)\int_{[a,b]}f(x)dx$.

1. $x$ 是有理数时，$D(x) = 1$
2. $x$ 是无理数时，$D(x) = 0$.

Dirichlet 函数是定义在实数轴的函数，并且值域是 $\{0,1\}$，无法画出函数图像，它不是 Riemann 可积的，但是它 Lebesgue 可积。

## 时间序列

### 时间序列的表示 — 基于 Riemann 积分

1. 分段线性逼近（Piecewise Linear Approximation）
2. 分段聚合逼近（Piecewise Aggregate Approximation）
3. 分段常数逼近（Piecewise Constant Approximation）

#### 分段聚合逼近（Piecewise Aggregate Approximation）— 类似 Riemann 积分

$\overline{x}_{i} = \frac{w}{N} \cdot \sum_{j=\frac{N}{w}(i-1)+1}^{\frac{N}{w}i} x_{j}$.

#### 符号特征（Symbolic Approximation）— 类似用简单函数来计算 Lebesgue 积分

SAX 方法的流程如下：

1. 正规化（normalization）：把原始的时间序列映射到一个新的时间序列，新的时间序列满足均值为零，方差为一的条件。
2. 分段表示（PAA）：$\{x_{1},\cdots, x_{N}\} \Rightarrow \{\overline{x}_{1},\cdots,\overline{x}_{w}\}$
3. 符号表示（SAX）：如果 $\overline{x}_{i}，那么 $\hat{X}_{i}=l_{1}$；如果 $z_{(j-1)/\alpha}\leq \overline{x}_{i}，那么 $\hat{X}_{i} = l_{j}$，在这里 $2\leq j\leq \alpha$；如果 $\overline{x}_{i}\geq z_{(\alpha-1)/\alpha}$，那么 $\hat{X}_{i} = l_{\alpha}$

### 时间序列的表示 — 基于 Lebesgue 积分

#### 熵（Entropy）

$\text{entropy}(X) = -\sum_{i=1}^{\infty}P\{x=x_{i}\}\ln(P\{x=x_{i}\})$.

#### 分桶熵（Binned Entropy）

$\text{binned entropy}(X) = -\sum_{k=0}^{\min(maxbin, len(X))} p_{k}\ln(p_{k})\cdot 1_{(p_{k}>0)},$

# 时序数据与事件的关联分析

### 关联关系的挖掘分成三个部分：

（1）是否存在关联性（Existence of Dependency）：在事件（E）与时间序列（S）之间是否存在关联关系。

（2）关联关系的因果关系（Temporal Order of Dependency）：是事件（E）导致了时间序列（S）的变化还是时间序列（S）导致了事件（E）的发生。

（3）关联关系的单调性影响（Monotonic Effect of Dependency）：用于判断时间序列（S）是发生了突增或者是突降。

### 基本概念：

$e_{i}$来表示某个事件，$\ell_{k}^{rear}(S,e_{i})$表示序列S在事件$e_{i}$之后的长度为k的子序列，$\ell_{k}^{front}(S,e_{i})$表示序列S在事件$e_{i}$之前的长度为k的子序列。如果事件E与时间序列S之间存在关联关系，那么

$\Gamma^{front}=\{\ell_{k}^{front}(S,e_{i}), i=1,\cdots,n\}$

$\Gamma^{rear}=\{\ell_{k}^{rear}(S,e_{i}),i=1,\cdots,n\}$应该是不一样的。

### 方法论：

$\Gamma^{front}$来做例子，$\Gamma^{front}=\{\ell_{k}^{front}(S,e_{i}), i=1,\cdots,n\}$$\Theta =\{\theta_{1},\cdots,\theta_{\tilde{n}}\}$ 是随机选择的，$Z=\Gamma \cup \Theta$，可以标记为$Z_{1},\cdots,Z_{p}$，其中$p=n$+$\tilde{n}$$Z_{i}=\ell_{k}^{front}(S,e_{i})$ when $1\leq i\leq n$$Z_{i}=\theta_{i-n}$ when $n$+$1\leq i\leq p$。可以使用记号$A=A_{1}\cup A_{2}$，其中$A_{1}=\Gamma^{front}$$A_{2}=\Theta=\{\theta_{1},\cdots,\theta_{\tilde{n}}\}$是随机选择的。

$I_{r}(x,A_{1},A_{2})=1$ when $x\in A_{i} \&\& NN_{r}(x,A)\in A_{i}$,

$I_{r}(x,A_{1},A_{2})=0$ when otherwise.

$T_{r,p}=\frac{1}{pr}\sum_{i=1}^{p}\sum_{j=1}^{r}I_{j}(x_{i},A_{1},A_{2})$,

$\lambda_{1}=n/p=n/(n$+$\tilde{n})$, $\lambda_{2}=\tilde{n}/(n$+$\tilde{n})$

$\alpha = 1.96$ for $P=0.025$

$\alpha = 2.58$ for $P=0.001$

$t_{score}=\frac{\mu_{\Gamma^{front}} - \mu_{\Gamma^{rear}}}{\sqrt{\frac{\sigma_{\Gamma^{front}}^{2}+\sigma_{\Gamma^{rear}}^{2}}{n}}}$.

$\alpha = 1.96$ for $P=0.025$

$\alpha = 2.58$ for $P=0.001$

### 算法综述：

7-13行是 $E\rightarrow S$ 的情形，因为$\Gamma^{rear}$ 异常，同时 $\Gamma^{front}$ 正常，说明事件导致了时间序列的变化。7-13行是为了计算 $t_{score}$ 的范围，判断是显著的提升还是下降。

14-20行是 $S\rightarrow E$ 的情形，因为$\Gamma^{front}$ 异常，就导致了事件的发生。14-20行是为了计算 $t_{score}$ 的范围，判断是显著的提升还是下降。

（1）Pearson Correlation

（2）J-Measure Correlation

# Opprentice: Towards Practical and Automatic Anomaly Detection Through Machine Learning

### 系统遇到的挑战：

Definition Challenges: it is difficult to precisely define anomalies in reality.（在现实环境下很难精确的给出异常的定义）

Detector Challenges: In order to provide a reasonable detection accuracy, selecting the most suitable detector requires both the algorithm expertise and the domain knowledge about the given service KPI (Key Performance Indicators). To address the definition challenge and the detector challenge, we advocate for using supervised machine learning techniques. （使用有监督学习的方法来解决这个问题）

### 该系统的优势：

(i) Opprentice is the first detection framework to apply machine learning to acquiring realistic anomaly definitions and automatically combining and tuning diverse detectors to satisfy operators’ accuracy preference.

(ii) Opprentice addresses a few challenges in applying machine learning to such a problem: labeling overhead, infrequent anomalies, class imbalance, and irrelevant and redundant features.

(iii) Opprentice can automatically satisfy or approximate a reasonable accuracy preference (recall>=0.66 & precision>=0.66). （准确率和覆盖率的效果）

### 2. 背景描述：

KPIs and KPI Anomalies:

KPIs: The KPI data are the time series data with the format of (time stamp, value). In this paper, Opprentice pays attention to three kinds of KPIs: the search page view (PV), which is the number of successfully served queries; The number of slow responses of search data centers (#SR); The 80th percentile of search response time (SRT).

Anomalies: KPI time series data can also present several unexpected patterns (e.g. jitters, slow ramp ups, sudden spikes and dips) in different severity levels, such as a sudden drop by 20% or 50%.

### 问题和目标：

1-FDR（false discovery rate）：# of false anomalous points detected / # of anomalous points detected = 1 – precision

The quantitative goal of opprentice is precision>=0.66 and recall>=0.66.

The qualitative goal of opprentice is automatic enough so that the operators would not be involved in selecting and combining suitable detectors, or tuning them.

### 3. Opprentice Overview: （Opprentice系统的概况）

(i) Opprentice approaches the above problem through supervised machine learning.

(ii) Features of the data are the results of the detectors.（Basic Detectors 来计算出特征）

(iii) The labels of the data are from operators’ experience.（人工打标签）

(iv) Addressing Challenges in Machine Learning: （机器学习遇到的挑战）

(1) Label Overhead: Opprentice has a dedicated labeling tool with a simple and convenient interaction interface. （标签的获取）

(2) Incomplete Anomaly Cases:（异常情况的不完全信息）

(3) Class Imbalance Problem: （正负样本比例不均衡）

(4) Irrelevant and Redundant Features:（无关和多余的特征）

### 4. Opprentice’s Design:

Architecture: Operators label the data and numerous detectors functions are feature extractors for the data.

Label Tool:

Detectors:

(i) Detectors As Feature Extractors: （Detector用来提取特征）

Here for each parameter detector, we sample their parameters so that we can obtain several fixed detectors, and a detector with specific sampled parameters a (detector) configuration. Thus a configuration acts as a feature extractor:

data point + configuration (detector + sample parameters) -> feature,

(ii) Choosing Detectors: (Detector的选择，目前有14种较为常见的）

Opprentice can find suitable ones from broadly selected detectors, and achieve a relatively high accuracy. Here, we implement 14 widely-used detectors in Opprentice.

Opprentice has 14 widely-used detectors:

Diff“: it simply measures anomaly severity using the differences between the current point and the point of last slot, the point of last day, and the point of last week.

MA of diff“: it measures severity using the moving average of the difference between current point and the point of last slot.

The other 12 detectors come from previous literature. Among these detectors, there are two variants of detectors using MAD (Median Absolute Deviation) around the median, instead of the standard deviation around the mean, to measure anomaly severity.

(iii) Sampling Parameters: （Detector的参数选择方法，一种是扫描参数空间，另外一种是选择最佳的参数）

Two methods to sample the parameters of detectors.

(1) The first one is to sweep the parameter space. For example, in EWMA, we can choose $\alpha \in \{0.1,0.3,0.5,0.7,0.9\}$ to obtain 5 typical features from EWMA; Holt-Winters has three [0,1] valued parameters $\alpha,\beta,\gamma$. To choose $\alpha,\beta,\gamma \in \{0.2,0.4,0.6,0.8\}$, we have $4^3$ features; In ARIMA, we can estimate their “best” parameters from the data, and generate only one set of parameters, or one configuration for each detector.

Supervised Machine Learning Models:

Decision Trees, logistic regression, linear support vector machines (SVMs), and naive Bayes. 下面是决策树（Decision Tree）的一个简单例子。

Random Forest is an ensemble classifier using many decision trees. It main principle is that a group of weak learners (e.g. individual decision trees) can together form a strong learner. To grow different trees, a random forest adds some elements or randomness. First, each tree is trained on subsets sampled from the original training set. Second, instead of evaluating all the features at each level, the trees only consider a random subset of the features each time. The random forest combines those trees by majority vote. The above properties of randomness and ensemble make random forest more robust to noises and perform better when faced with irrelevant and redundant features than decisions trees.

Configuring cThlds: （阈值的计算和预估）

(i) methods to select proper cThlds: offline part

We need to figure cThlds rather than using the default one (e.g. 0.5) for two reasons.

(1) First, when faced with imbalanced data (anomalous data points are much less frequent than normal ones in data sets), machine learning algorithems typically fail to identify the anomalies (low recall) if using the default cThlds (e.g. 0.5).

(2) Second, operators have their own preference regarding the precision and recall of anomaly detection.

The metric to evaluate the precision and recall are:

(1) F-Score: F-Score = 2*precision*recall/(precision+recall).

(2) SD(1,1): it selects the point with the shortest Euclidean distance to the upper right corner where the precision and the recall are both perfect.

(3) PC-Score: （本文中采用这种评估指标来选择合适的阈值）

If r>=R and p>=P, then PC-Score(r,p)=2*r*p/(r+p) + 1; else PC-Score(r,p)=2*r*p/(r+p). Here, R and P are from the operators’ preference “recall>=R and precision>=P”. Since the F-Score is no more than 1, then we can choose the cThld corresponding to the point with the largest PC-Score.

(ii) EWMA Based cThld Prediction: （基于EWMA方法的阈值预估算法）

In online detection, we need to predict cThlds for detecting future data.

Use EWMA to predict the cThld of the i-th week ( or the i-th test set) based on the historical best cThlds. Specially, EWMA works as follows:

If $i=1$, then $cThld_{i}^{p}=cThld_{1}^{p}=$ 5-fold prediction

Else $i>1$, then $cThld_{i}^{p}=\alpha\cdot cThld_{i-1}^{b}$+$(1-\alpha)\cdot cThld_{i-1}^{p}$, where $cThld_{i-1}^{b}$ is the best cThld of the (i-1)-th week. $cThld_{i}^{p}$ is the predicted cThld of the i-th week, and also the one used for detecting the i-th week data. $\alpha\in [0,1]$ is the smoothing constant.

For the first week, we use 5-fold cross-validation to initialize $cThld_{1}^{p}$. As $\alpha$ increases, EWMA gives the recent best cThlds more influences in the prediction. We use $\alpha=0.8$ in this paper.

### 5. Evaluation（系统评估）

Opprentice has 14 detectors with about 9500 lines of Python, R and C++ code. The machine learning block is based on the scikit-learn library.

Random Forest is better than decision trees, logistic regression, linear support vector machines (SVMs), and naive Bayes.

# Focus: Shedding Light on the High Search Response Time in the Wild

### 问题描述：

To help search operators dubug HSRT (high search response time)，Focus is a search log analysis framework to answer the three questions:

(1) What is the HSRT condition?

(2) Which HSRT condition types are prevalent across days?

(3) How does each attribute affect SRT in those prevalent HSRT condition types?

### 解决方案：

Focus has one component for each of the above questions:

(1) A decision tree based classifier to identify HSRT conditions in search logs of each day;

(2) A clustering based condition type miner to combine similar HSRT conditions into one type, and find the prevalent condition types across days; following Occam’s razor principle.

(3) An attribute effect estimator to analyze the effect of each individual attribute of SRT within a prevalent condition type.

### 基础知识准备：

(A) Search Logs:

For each measured query, its search log records two types of data: SRT and SRT components, Query Attributes.

(1) SRT and SRT components:（特征层）

$t_{1}$ is when a query is submitted; $t_{2}$ is when the result HTML file has been downloaded; $t_{3}$ is when a brower finishes parsing the HTML; $t_{4}$ is when the page is completely rendered. SRT is measured by $t_{4}-t_{1}$, the user-received search response time.

$T_{server}$ is the server response time of the HTML file, which is recorded by servers; $T_{net}=t_{2}-t_{1}-T_{server}$ is the network transmission time of the HTML file; $T_{brower}=t_{3}-t_{2}$ is the browser parsing time of the HTML; $T_{other}=t_{4}-t_{3}$ is the remaining time spent before the page is rendered, e.g. download time of images from image servers.

(2) Query Attributes:（特征层）

The search logs record the following attributes for each measured query:

(i) Browser Engine: Webkit(e.g. Chrome, Safari and 360 Secure Browser), Gecko, Trident LEGC, Trident 4.0, Trident 5.0, and others.

(ii) ISP: China Telecom, China Unicom, China Mobile, China Netcom, CHina Tietong, others.

(iii) Localtion: Based on the client IP, convert IP to its geographic location. In total, there are 32 provinces.

(iv) #Image: the number of embedded images in the result page.

(vii) Background page views: On the service side, the search engine S also post-analyzes the logs and generates the background page views. The background PVs (page views) for a query q is measured by the number of queries served within 30 seconds before and after q is served.It reflects the average search request load where q is served. Due to confidentiality constraints, we normalize specific background PVs (page Views) by the maximum value.（事后分析，统计出一些必要的特征，输入 Focus 系统的机器学习模型中）

(B) HSRT and HSRT Conditions:（样本层）

Usually, we can use cumulative distribution fraction (CDF) of SRT in the search logs to determine the high search response time condition (HSRT condition). In this paper,  we define HSRT as the SRT longer than 1s.

Challenges of Identifying HSRT Conditions: In order to identify HSRT conditions in multi-dimensional search logs.（以下是这个系统的一些难点和挑战点）

(a) Naive Single Dimensional Based Methods: including pair-wise correlation analysis and so on, but is inefficient.

(b) Attributes can be potentially interdependent on each other: that means Naive Bayes Method may not applicable in this situation.

(c) Need to avoid output overlapping conditions: like {#image>30}, {ads=yes}, and {#image>20, ads=yes}.  （随着时间的推移，每天使用模型可能会推出类似或者重复的规则）。

### 关键思想和系统概况

Condition is a combination of attributes and specific values in search logs.

HSRT Condition is a condition that covers at least 1%\$ of total queries, and has the fraction of HSRT large than the global level:

(# of HSRT queries in a HSRT condition / #of queries in a HSRT condition) > (# of HSRT queries / # of queries). This is in order to assign to labels and we can change this definition in practice. （这只是用来打标签的定义，用于判断什么是HSRT，在实际的应用中，我们可以根据具体的场景采用不同的定义，例如返回码等指标）。

### ‘Focus’ System Overview:

Input: search logs（日志）

(i) Use a decision tree based classifier to identify HSRT conditions  in search logs every day; （每天可以使用决策树模型从日志中提取HSRT条件）。

(ii) Use a clustering based condition type miner to identify condition types of similar HSRT conditions, and fine prevalent condition types across days; （用于把类似的条件融合在一起）。

(iii) Use an attribute effect estimator to analyze how an attribute affects SRT and SRT components in each prevalent condition type. （用于判断哪些属性或者特征对这个标签影响更加深远）。

Output: prevalent condition types and their attributes effects on SRT.（第二步输出的条件以及第三步属性的重要性）。

Part (i): Decision Tree Based Classifier including ID3, C4.5, CART. It contains five important parts: (1) expressing attribute splits; (2) evaluate splits; (3) stopping tree growing; (4) assigning Labels: assign HSRT labels to the left nodes whose fraction of HSRT is larger than the global fraction of HSRT; (5) identify HSRT Branching Attribute Conditions. （这里是 Focus 系统所采用的机器学习算法）。

Part (ii): Condition Type Miner: group HSRT conditions according to (1) the same combination of attributes, (2) the same value from each category attribute, and (3) similar interval for each numeric attribute, using Jaccard Index to measure the similarity between intervals. （条件的融合）。

Part (iii): Attribute Effect Estimator: With each condition type

$C=\{c_{1}\wedge c_{2}\cdots \wedge c_{i} \wedge \cdots c_{n}\}$,

we design a method to understand how each attribute condition $c_{i}$ affects SRT.

For example, what is the HSRT fraction caused by $c_{i}$ in $C$? What SRT components (e.g. $T_{net}$ and $T_{server}$) are affected by $c_{i}$?

Main Idea: flip condition $c_{i}$ to the opposite $\overline{c}_{i}$ to get a variant condition type $C_{i}'=\{c_{1}\wedge c_{2}\cdots \wedge \overline{c}_{i} \wedge \cdots c_{n}\}$. In the past days, we have the number of HSRT events in total, the number of HSRT events in condition $C$ and the number of HSRT events in condition $C_{i}'$. As a result, we believe the historical data based comparison can provide a reasonable estimate of the attribute effects. The comparison between $C$ and $C_{i}'$ in these days is based on the specific HSRT conditions of these days. （用于判断哪些属性更能够引起 HSRT）。

In Table IV, the results are sorted by the variation of the fraction of HSRT in condition types (HSRT% column) caused by flipping an attribute condition.

(i) We highlight the variations greater than zero (getting worse after flipping an attribute condition).

(ii) We focus on that flipping the HSRT branching attribute conditions can yield improvements on HSRT%. For example, the condition #image>x are all ranked at the top. It means we need to reduce the impact of images on SRT and we can get the highest potential improvement of HSRT.

(iii) Table III and Table IV are the output of Focus to the operators for these months.

### Observations by Further Inverstigation

Table IV raises some interesting questions:（通过 Focus 输出的表格 Table IV 可以提出很多其余的问题，也许是人工经验不容易发现的问题）

(1) Why does reducing #images increase $T_{server}$, the time that servers prepare the result HTML (row 1, 2, 3, 4 of Table IV)?

(2) How do ads inflate SRT? Why do the pages with ads need more $T_{net}$ and $T_{brower}$ (row 7)?

(3) Why does Webkit engine perform better, especially greatly decreasing $T_{browser}$ (row 5, 10, 11, 12)?

(4) It is nature that switching ISPs can affect network transmission time $T_{net}$, but why does switching to China Telecom reduce $T_{server}$ by over 20% (row 6, 8, 9)?